/sci/ - Science & Math

I'll repost this from some other thread.

I'm currently working on problem (e) of pic related and the proof strategies I'm coming up with for it are ugly as hell, tedious, cumbersome and rather long. They furthermore barely involve the result of (a) and (c), which makes me rather suspicious.

I do not want an answer to this problem but merely to know if you guys can actually come up with a short and elegant proof for this that doesn't involve tedious construction of families of intervals. Don't tell me the answer if you have it, just tell me whether it's long or short.

Can any math fag help me out with this? I don't actually want an answer for this problem but merely want to know how long your proof for it is, or if the solution is obvious to you / elegant. I believe I might have found a way to solve this but it basically takes a page or more and I have to restrict myself, in parts, to disjoint open intervals and generalize afterward, which I'm not even sure works since my strategy also implicitly relies on the idea that all [math]\mathcal{I_j}[\math] are subsets of [math][a,b][\math], which does not appear to correct.

The fact that I'm not relying more on (a) an (c) also seems to me to be highly suspicious.

So, are you guys able to prove this in a few lines or in an elegant manner or is your proof also tedious as hell?